EE354 : Communications System I Lecture 2,3: Signals in communication systems Fourier review Aliazam Abbasfar

Outline Signals Fourier Series Fourier Transform Fourier properties Linear systems Channel model

Signals in communication systems m(t) m[n] m(t) m[n] x(t) y(t) message Source encoder Transmitter Source decoder Channel Receiver Analog systems m(t) is a continuous signal Digital systems m[n] is a discrete signal m[n] takes limited values x(t) t x(t) T t

Signals : Important parameters Energy, power Frequency components DC level Bandwidth Power spectral density

Energy and Power Signals x(t) is an energy signal if E is finite x(t) is an power signal if P is finite Energy signals have zero power Power signals have infinite energy

Tone signals Single tone signal Multi-tone signal Periodic with period T0 Frequency content only at f0 Amplitude and phase = phasor One-sided/Two-sided spectrum We show the spectrum with respect to f ( NOT w) Power = A2/2 Multi-tone signal Bandwidth

Fourier series Periodic signals with period T0 f0 = 1/T0 : fundamental frequency cn :Line(discrete) spectrum of the signal Parseval’s theorem :

Fourier Transform Continuous spectrum Real signals : X(-f) = X*(f) Even signals : X(f) is real Odd signals : X(f) is imaginary

Rectangular pulse Rect(t) : a pulse with unit amplitude and width Sinc(f) = sin(pf)/(pf) Band-limited and time-limited signals

Fourier Transform Properties Useful properties Linearity Time shift Time/Freq. scaling Modulation Convolution/multiplication Differentiation/integration Duality:  Parseval’s equation : Energy and energy spectral density

Special signals DC x(t) = 1  X(f) = d(f) Impulse x(t) = d(t)  X(f) = 1 Sign x(t) = sgn(t)  X(f) = 1/jpf Step x(t) = u(t)  X(f) = 1/j2pf+ 1/2d(f) Tone x(t) = ej2pf0t  X(f) = d(f-f0) Periodic signals

Fourier examples Impulse train: x(t) = Sd (t-nT0)  X(f) = 1/T0Sd (f-nf0) Repetition y(t) = repT(x) = S x(t-nT)  Y(f) = 1/T S X(n/T) d (f-n/T) Sampling y(t) = combT(x) = S x(nT) d (t-nT)  Y(f) = 1/T S X(f-n/T)

Fourier Transform and LTI systems An LTI system is defined by its impulse response, h(t) H(f) : frequency response of system x(t) = ej2pfot  y(t) = H(f0) ej2pfot Eigen-functions and Eigen-values of any LTI system Bandwidth

Channel model Channels are often modeled as LTI systems h(t) : channel impulse response H(f) : channel frequency response Noise is added at the receiver Additive noise Lowpass and passband channels Channel bandwidth

Power measurement PdBW = 10 log10(P/1 W) PdBm = 10 log10(P/1 mW) = PdBW + 30 Power gain g = Pout/Pin gdB = 10 log10( Pout/Pin) Power loss L = 1/g = Pin/Pout LdB = 10 log10( Pin/Pout) Transmission gain Pout = g1g2g3g4 Pin= g2g4 /L1L3 Pin in dB : Pout = g1 + g2 + g3 +g4 + Pin= g2 + g4 - L1 – L3 + Pin

Reading Carlson Ch. 2 and 3.1 Proakis 2.1, 2.2